Bivariate Exponential Distributions Using Linear Structures

We derive bivariate exponential distributions using independent auxiliary random variables. We develop separate models for positive and negative correlations between the exponentially distributed variates. To obtain a positive correlation, we define a linear relation between the variates X and Y of the form Y = aX +Z where a is a positive constant and Z is independent of X. To obtain exponential marginals for X and Y we show that Z is a product of a Bernoulli and an Exponential random variables. To obtain negative correlations, we define X = aP +V and Y = bQ+W where either P and Q or V and W or both are antithetic random variables. For the case of positive correlations, we also characterize a bivariate Poisson process generated by using the bivariate exponential as the interarrival distribution. Our bivariate exponetial model is useful in introducing dependence between the interarrivals and service times in a queueing model and in the failure process in multicomponent systems. The primary advantage of our model is that the resulting queueing and reliability analysis remains mathematically tractable because the Laplace Transform of the joint distribution is a ratio of polynomials of s. Further, the variates can be very easily generated for computer simulation.